Set up a charcoal fire for direct grilling. Put a cast iron skillet in the hot coals.
In a small saucepan, melt 3 tablespoons of the butter. Dredge the trout fillets through the melted butter and season each with salt and pepper.
Put the trout fillets, flesh side down, in the hot skillet and cook for 1 1/2 minutes. Flip the fillets and cook, skin side down, until the fish starts to flake with pressure, 1 1/2 minutes. Remove the pan from the coals and set it on a non-flammable surface. Transfer the trout to a plate.
Wait 3 minutes for the pan to cool slightly. Add the remaining 6 tablespoons of butter and cook until the butter starts to brown, about 30 seconds.
Add the sun-dried tomatoes, capers, and parsley to the pan and cook, stirring, for 30 seconds.
Pour in the white wine and lemon juice and stir, scraping up any bits from the bottom of the pan, for about 1 minute. Simmer until the sauce has reduced by half. Season with salt and pepper, if desired.
Drizzle the sauce over the trout fillets and serve.